Differential equation- wordy question1- urgent

6. The number of people, x, in a queue at a travel centre t minutes after it opens is modelled by a differential equation dx/dt= 1.4t-0.5x for values of t up to 10. Sovlve the differential equation given that x=8 when t=o.

b. An alternative model gives the differential equation dx/dt= 1.4t-o.5x for the same values of t. Verify that x=13.6e^-0.5t+2.8t-5.6 satisfies this differential equation. Verify also that when t=0 this function takes the value 8.

6. The number of people, x, in a queue at a travel centre t minutes after it opens is modelled by a differential equation dx/dt= 1.4t-0.5x for values of t up to 10. Sovlve the differential equation given that x=8 when t=o.

You have:

with initial condition when .

Now rewrite the DE as:

and we see that we have an inhomogeneous linear ODE with constant
coefficients, and so a general solution is the sum of the general solution
of the homogeneous equation:

,

and any particular integral of the original equation (1).

Now the homogeneous equation (2) can be solved by a number of methods,
we can use a trial solution , substitute this into
the equation and find the value of that gives the correct answer.
Another method is to note that (2) is of variable separable type.

I will use the first of these methods, I will suppose that: , then:

,

so if this is a solution we must have , and the general
solution of (2) is: .

Now we need to find a particular integral for (1). Here we note that the RHS is
a multiple of , so if the LHS can be made to equal
the RHS with suitable choice of and , so substituting this into (1) we get:

.

So equating the coefficient of in the above gives , so ,
and the constant terms gives , or .

Hence a particular integral of (1) is:

.

Combining this with the general solution of (2) found earlier we get the
general solution of (1) is:

An alternative model gives the differential equation dx/dt= 1.4t-o.5x for the same values of t. Verify that x=13.6e^-0.5t+2.8t-5.6 satisfies this differential equation. Verify also that when t=0 this function takes the value 8.

Do the last part first, substitute into the equation:

to get:

,

as required.

Now to verify that (A) satisfies the DE we differentiate , and check
that this derivative equals , which I leave as an exercise for the reader.